Simplify; express your answer in exponential form. Assume $k\neq 0, x\neq 0$. $\dfrac{{(k^{4}x^{5})^{-5}}}{{(k^{5}x^{3})^{-3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{4}x^{5})^{-5} = (k^{4})^{-5}(x^{5})^{-5}}$ On the left, we have ${k^{4}}$ to the exponent ${-5}$ . Now ${4 \times -5 = -20}$ , so ${(k^{4})^{-5} = k^{-20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{4}x^{5})^{-5}}}{{(k^{5}x^{3})^{-3}}} = \dfrac{{k^{-20}x^{-25}}}{{k^{-15}x^{-9}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-20}x^{-25}}}{{k^{-15}x^{-9}}} = \dfrac{{k^{-20}}}{{k^{-15}}} \cdot \dfrac{{x^{-25}}}{{x^{-9}}} = k^{{-20} - {(-15)}} \cdot x^{{-25} - {(-9)}} = k^{-5}x^{-16}$